\newproblem{lay:1_6_7}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.6.7}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  Alka-Seltzer contains sodium bicarbonate ($NaHCO_3$) and citric acid ($H_3C_6H_5O_7$). When a tablet is dissolved in water,
	the following reaction produces sodium citrate, water and carbon dioxide (gas):
	\begin{center}
		$NaHCO_3+H_3C_6H_5O_7 \rightarrow Na_3C_6H_5O_7+H_20+CO_2$
	\end{center}
	Balance this chemical equation.
}{
   % Solution
	Let's assign a number of molecules to each one of the compounds
	\begin{center}
		$x_1NaHCO_3+x_2H_3C_6H_5O_7 \rightarrow x_3Na_3C_6H_5O_7+x_4H_2O+x_5CO_2$
	\end{center}
	Now let's count the number of atoms of each kind
	\begin{center}
		\begin{tabular}{lc}
			$Na$: & $x_1=3x_3$ \\
			$H$:  & $x_1+3x_2+5x_2=5x_3+2x_4$ \\
			$C$:  & $x_1+6x_2=6x_3+x_5$ \\
			$O$:  & $3x_1+7x_2=7x_3+x_4+2x_5$ \\
		\end{tabular}
	\end{center}
	The augmented matrix of this equation system is
	\begin{center}
		$\left(\begin{array}{rrrrr|r}
		   1 & 0 & -3 &  0 &  0 & 0 \\
			 1 & 8 & -5 & -2 &  0 & 0 \\
			 1 & 6 & -6 &  0 & -1 & 0 \\
			 3 & 7 & -7 & -1 & -2 & 0
		 \end{array}\right)\sim
		 \left(\begin{array}{rrrrr|r}
		   1 & 0 &  0 &  0 & -1 & 0 \\
			 0 & 1 &  0 &  0 & -\frac{1}{3} & 0 \\
			 0 & 0 &  1 &  0 & -\frac{1}{3} & 0 \\
		   0 & 0 &  0 &  1 & -1 & 0 \\
		 \end{array}\right)$
	\end{center}
	Letting $x_5=3$, we have $x_1=x_5=3$, $x_2=\frac{1}{3}x_5=1$, $x_3=\frac{1}{3}x_5=1$, $x_4=x_5=3$. Finally, the balanced chemical reaction is
	\begin{center}
		$3NaHCO_3+H_3C_6H_5O_7 \rightarrow Na_3C_6H_5O_7+3H_2O+3CO_2$
	\end{center}
}
\useproblem{lay:1_6_7}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
